Trigonometry MCQ1) Which of the following is the correct value of cot 100.cot 200.cot 600.cot 700.cot 800?
Answer: (a) 1/√3 Explanation: Here, we can apply the formula - cot A. cot B = 1 (when A + B = 900) = (cot 200 . cot 700) x (cot 100 . cot 800) x cot 600 = 1 x 1 x 1/√3 = 1/√3 So, the correct value of cot 100.cot 200.cot 600.cot 700.cot 800 = 1/√3 2) If a sin 450 = b cosec 300, what is the value of a4/b4?
Answer: (b) 43 Explanation: Given a sin 450 = b cosec 300 So, a/b = cosec 300/ sin 450 a/b = 2/( 1/√2) a/b = 2√2/1 a4/b4 = (2√2/1)4 a4/b4 = 64/1 or, a4/b4 = 43 3) If tan θ + cot θ = 2, then what is the value of tan100 θ + cot100 θ?
Answer: (c) 2 Explanation: Given tan θ + cot θ = 2 Put θ = 450, above equation will satisfy as, 1 + 1 = 2 So, θ = 450, = tan100 450 + cot100 450 = 1100 + 1100 = 2 4) If the value of α + β = 900, and α : β = 2 : 1, then what is the ratio of cos α to cos β ?
Answer: (c) 1 : √3 Explanation: Given α + β = 900, and α : β = 2 : 1 So, we can say that 2x + x = 900 3x = 900, which give x = 300 So, α = 2x = 60 β = x = 30 cos α / cos β = cos 600 / cos 300 => (1/2) / (√3/2) or, 1/2 * 2/√3 = 1/√3 Or the ratio between cos α : cos β = 1 : √3 5) If θ is said to be an acute angle, and 7 sin2 θ + 3 cos2 θ = 4, then what is the value of tan θ?
Answer: (c) 1/√3 Explanation: Given 7 sin2 θ + 3 cos2 θ = 4 => 7 sin2 θ + 3 (1 - sin2 θ) = 4 => 7 sin2 θ + 3 - 3sin2 θ = 4 Then, 4sin2 θ = 1 Or, sin θ = 1/2 So, θ = 300 Now, put θ = 300 in tan θ, we will get, tan θ = 1/√3 Alternate We can directly check the equation by putting values of θ. Let's put θ = 300 7 sin2 300 + 3 cos2 300= 4 Then, 7 * 1/4 + 3 * 3/4 = 4 So, 7/4 + 9/4 = 4 16/4 = 4 Or, 4 = 4 (so, it satisfy the condition) Now, tan 300 = 1/√3 6) If tan θ - cot θ = 0, what will be the value of sin θ + cos θ?
Answer: (b) √2 Explanation: Given tan θ - cot θ = 0 Let's put θ = 450 in order to satisfy the above equation tan 450 - cot 450 = 0 1 - 1 = 0 (equation satisfied with θ = 450) Now, put θ = 450 in sin θ + cos θ, we will get = sin 450 + cos 450 = 1/√2 + 1/√2 = √2 7) If θ is said to be an acute angle, and 4 cos2 θ - 1 = 0, then what is the value of tan (θ - 150)?
Answer: (a) 1 Explanation: Given 4 cos2 θ - 1 = 0 4 cos2 θ = 1 cos2 θ = 1/4 cos θ = 1/2 Or, θ = 600 So, tan (θ - 150) = ? => tan (600 - 150) = tan 450 = 1 8) If the value of θ + φ = π/2, and sin θ = 1/2, what will be the value of sinφ?
Answer: (c) √3/2 Explanation: Given θ + φ = π/2 It can be written as, θ + φ = 900 (as π = 1800) …….(i) sin θ = 1/2 or, θ = 300 On putting the value of θ = 300 in equation (i), we will get, 300 + φ = 900 So, φ = 600 Then, sin φ = sin 600 = √3/2 9) What will be the value of 2cos2 θ - 1, if cos4 θ - sin4 θ = 2/3?
Answer: (d) 2/3 Explanation: Given cos4 θ - sin4 θ = 2/3 Now, here we can apply the formula - a4 - b4 = (a2 - b2) (a2 + b2) So, (cos2 θ - sin2 θ) (cos2 θ + sin2 θ) = 2/3 So, 1 x (cos2 θ - sin2 θ) = 2/3 (because cos2 θ + sin2 θ = 1) => cos2 θ - (1 - cos2 θ) = 2/3 (because sin2 θ = 1 - cos2 θ) So, 2cos2 θ - 1 = 2/3 10) What will be the value of 1 - 2sin2 θ, if cos4 θ - sin4 θ = 2/3?
Answer: (d) 2/3 Explanation: Given cos4 θ - sin4 θ = 2/3 Now, here we can apply the formula - a4 - b4 = (a2 - b2) (a2 + b2) So, (cos2 θ - sin2 θ) (cos2 θ + sin2 θ) = 2/3 So, 1 x (cos2 θ - sin2 θ) = 2/3 (because cos2 θ + sin2 θ = 1) => (1 - sin2 θ) - sin2 θ = 2/3 So, 1 - 2sin2 θ = 2/3 11) What is the value of tan θ/(1 - cot θ) + cot θ/(1 - tan θ)?
Answer: (a) tan θ + cot θ + 1 Explanation: tan θ/(1 - (1/tan θ) + (1/tan θ)/(1 - tan θ) = tan2 θ/ (tan θ - 1) - 1/tan θ(tan θ - 1) = tan3 θ - 1/tan θ(tan θ - 1) Apply the formula, a3- b3 = (a - b) (a2 + ab + b2) = (tan θ - 1) (tan2 θ + tan θ + 1) / tan θ(tan θ - 1) = (tan2 θ + tan θ + 1) / tan θ On taking tan θ common from the numerator, we will get, = tan θ + cot θ + 1 12) What is the value of sin θ/(1 + cos θ) + sin θ/(1 - cos θ), where (00 < θ < 900)?
Answer: (a) 2cosec θ Explanation: Given, sin θ/(1 + cos θ) + sin θ/(1 - cos θ) = [sin θ (1 - cos θ) + sin θ (1 + cos θ)] / [(1 - cos θ) (1 + cos θ)] = [sin θ - sin θ cos θ + sin θ + sin θ cos θ] / [1 - cos2 θ] = 2 sin θ / sin2 θ = 2 cosec θ 13) What will be the value of (√3 tanθ + 1), if r sinθ = 1, and r cosθ = √3?
Answer: (a) 2 Explanation: Given, r sinθ = 1, and r cosθ = √3 r sinθ / r cosθ = 1/√3 tanθ = 1/√3 or √3 tanθ = 1 So, √3 tanθ + 1= 1 + 1 = 2 14) What is the value of (tan2 θ - sec2 θ)?
Answer: (b) -1 Explanation: (tan2 θ - sec2 θ) = sin2 θ/cos2 θ - 1/cos2 θ = (sin2 θ - 1) / cos2 θ = - cos2 θ/cos2 θ = -1 15) If sin θ = 0.7, then what is the value of cosθ, if 00 <= θ < 900?
Answer: (a) √0.51 Explanation: Given sin θ = 0.7 As we know, sin2 θ + cos 2 θ = 1 So, (0.7)2 + cos 2 θ = 1 Then, 0.49 + cos 2 θ = 1 => cos2 θ = 1 - 0.49 cos θ = √0.51 16) What is the value of tan3θ, If tan7θ.tan2θ = 1?
Answer: (b) 1/√3 Explanation: Given tan7θ.tan2θ = 1 As we know, if tanA . tanB = 1 then, A + B = 900 So, 7θ + 3θ = 900 => 9θ = 900 Or, θ = 100 Now, we have to find tan3θ So, put θ = 100 in tan3θ, we will get tan 300 = 1/√3 17) What will be the value of 3cos800.cosec100 + 2cos590.cosec310?
Answer: (c) 5 Explanation: 3cos800.cosec100 + 2cos590.cosec310 = ? According to the identity, [if A + B = 900 then, cosA.cosecB = 1] So, 3cos800.(1/sin100) + 2cos590.(1/sin310) = 3cos800.(1/sin(900 - 800)) + 2cos590.(1/sin(900 - 590)) => 3cos800/cos 800) + 2cos590/cos 590 ( because sin (900 - θ) = cos θ) = 3 + 2 = 5 18) If sin (θ + 180) = cos 600, then what is the value of cos5θ, where 00 < θ < 900?
Answer: (b) 1/2 Explanation: Given sin (θ + 180) = cos 600 sin (θ + 180) = cos (900 - 300) So, sin (θ + 180) = sin300 Then, θ = 300 - 180 θ = 120 So, cos5θ = cos 5 x 120 = cos 600 = 1/2 19) If cos A = 2/3, then what is the value of tan A?
Answer: (d) √5/2 Explanation: According to the trigonometric identities, 1 + tan2 A = sec2 A And we know, sec A = 1/cos A So, sec A = 1/(2/3) = 3/2 Then, 1 + tan2 A = (3/2)2 = 9/4 => tan2 A = 9/4 - 1 => tan2 A = 5/4 So, tan A = √5/2 20) What will be the simplified value of (sec A sec B + tan A tan B)2 - ( sec A tan B + tan A sec B)2?
Answer: (b) 1 Explanation: The question is in the form of (a + b)2 So, on applying the identity, and after expanding the given equation, we will get - => sec2 A sec2 B + tan2 A tan2 B + 2 sec A sec B tan A tan B - sec2 A tan2 B - tan2 A sec2 B - 2 sec A tan B tan A sec B => Then, sec2 A [sec2 B - tan2 B] - tan2 A [sec2 B - tan2 B] So, it will be written as [sec2 A - tan2 A] [sec2 B - tan2 B] = 1 x 1 = 1. 21) What is the simplified value of (cosec A - sin A)2 + (sec A - cos A)2 - (cot A - tan A)2?
Answer: (b) 1 Explanation: The question is in the form of (a - b)2 (a - b)2 = a2 + b2 - 2ab So, on applying the identity, and after expanding the given equation, we will get - => cosec2 A + sin2 A - 2 cosec A sin A + sec2 A + cos2 A - 2 sec A cos A - cot2 A - tan2 A + 2 cot A tan A After solving it with using trigonometric identities, we will get - => (cosec2 A - cot2 A) + (sin2 A + cos2 A) + (sec2 A - tan2 A) -2 = 1 + 1 + 1 - 2 = 3 - 2 = 1 Alternate method (cosec A - sin A)2 + (sec A - cos A)2 - (cot A - tan A)2 We can solve it directly by putting θ = 450 = (cosec 450 - sin 450)2 + (sec 450 - cos 450)2 - (cot 450 - tan 450)2 = (√2 - 1/√2)2 + (√2 - 1/√2)2 - (1 - 1)2 = 1/2 + 1/2 - 0 = 1 22) What will be the value of sec4 θ - tan4 θ, if sec2 θ + tan2 θ = 7/12?
Answer: (b) 7/12 Explanation: Given sec2 θ + tan2 θ = 7/12 Now, here we can apply the formula - a4 - b4 = (a2 - b2) (a2 + b2) sec4 θ - tan4 θ = (sec2 θ - tan2 θ) (sec2 θ + tan2 θ) = 1 x (sec2 θ + tan2 θ) {because 1 + tan2 θ = sec2 θ} = 1 x 7/12 = 7/12 23) What is the value of cos2 200 + cos2 700?
Answer: (c) 1 Explanation: cos2 200 + cos2 700 We can write cos2 200 as cos2 (900 - 700) So, cos2 (900 - 700) + cos2 700 = sin2 700 + cos2 700 = 1 24) If r cosθ = √3, and r sinθ = 1, what is the value of r2 tanθ?
Answer: (a) 4/√3 Explanation: Given r cosθ = √3, and r sinθ = 1 r cosθ / r sinθ = 1/√3 tanθ = tan 300 Or, θ = 300 On putting, θ = 300, we will get, r sin 300 = 1 r x ½ = 1 or r =2 Now, r2 tanθ = ? = (2)2 tan 300 = 4 x 1/√3 = 4/√3 25) Which of the following is the correct value of tan2 A + cot2 A - sec2 A cosec2 A, where 00 < A < 900?
Answer: (c) -2 Explanation: We can solve it by putting θ = 450 On putting θ = 450, we will get - = tan2 450 + cot2 450 - sec2 450 cosec2 450 = 1 + 1 - (√2)2 x (√2)2 = 2 - 4 = -2 26) If the value of tan2 θ + tan4 θ = 1, what will be the value of cos2 θ + cos4 θ?
Answer: (b) 1 Explanation: Given, tan2 θ + tan4 θ = 1 …. (i) From equation (i), tan2 θ ( 1 + tan2 θ ) = 1 tan2 θ ( sec2 θ ) = 1 [As according to the trigonometric identity, sec2 θ - tan2 θ = 1] tan2 θ = 1/ sec2 θ tan2 θ = cos2 θ ….(ii) Now, cos2 θ + cos4 θ = ? => cos2 θ + (cos2)2 θ => tan2 θ + (tan2)2 θ => tan2 θ + tan4 θ = 1 {from equation (i)} 27) If 4sin2 θ = 3, and θ is a positive acute angle, what is the value of tan θ - cot θ/2?
Answer: (b) 0 Explanation: Given, 4 sin2 θ = 3 Or, sin2 θ = 3/4 Or, sin θ = √3/2 So, θ = 600 Now, tan θ - cot θ/2 = ? Put θ = 600 = tan 600 - cot 600/2 = tan 600 - cot 300 = √3 - √3 = 0 28) If the value of sin A + cosec A = 2, then what is the value of sin7 A + cosec7 A?
Answer: (c) 2 Explanation: It is given that sin A + cosec A = 2 ……(i) On putting A = 900, then above condition will satisfy sin 900 + cosec 900 = 2 or, 1 + 1 = 2 (as the equation satisfies, so, A = 900) Now, sin7 A + cosec7 A = ? => sin7 900 + cosec7 900 => 17 + 17 = 2 29) What is the value of (2tan 300) / (1 + tan2 300)?
Answer: (c) sin 600 Explanation: tan 300 = 1/√3 (2tan 300) / (1 + tan2 300) = ? = (2 x 1/√3) / (1 + (1/√3)2) = (2/√3) / (4/3) = 6/4√3 = Or √3/2, which is equal to option C, i.e., sin600. 30) What is the value of (sin 300 + cos 600) - (sin 600 + cos 300)?
Answer: (c) 1 + √3 Explanation: Let's see the values - sin 300 = 1/2 cos 600 = 1/2 sin 600 = √3/2 cos 300 = √3/2 So, (1/2 + 1/2) - (√3/2 + √3/2) = 1 - 2√3/2 Or, 1 - √3 31) If 1 + cos2 θ is equal to 3 sin θ.cos θ, then what is the value of cot θ?
Answer: (a) 1 Explanation: It is given that, 1 + cos2 θ = 3 cos θ.sin θ On dividing both sides by sin2 θ, we will get 1+cos2 θ / sin2 θ = 3 cos θ.sin θ/sin2 θ cosec2 θ + cot2 θ = 3 cot θ => 1 + cot2 θ + cot2 θ = 3 cot θ [because 1 + cot2 θ = cosec2 θ] => 1 + 2cot2 θ = 3 cot θ => 2 cot2 θ = 3 cot θ - 1 Let's try to put θ = 450 => 2cot2 450 - 3 cot 450 + 1 = 0 => 2 -3 + 1 = 0 => 0 = 0 (satisfies) So, θ = 450 cot θ = cot 450 = 1 32) If the value of tan θ = 4/3, then which of the following is the correct value of (3 sin θ + 2 cos θ) / (3 sin θ - 2 cos θ) =?
Answer: (d) 3 Explanation: It is given that, tan θ = 4/3 => sin θ / cos θ = 4/3 So sin θ = 4, and cos θ = 3 Now, on putting the values of sin θ and cos θ in (3 sin θ + 2 cos θ) / (3 sin θ - 2 cos θ), we will get - = 3x4 + 2x3/ 3x4 - 2x3 = 18/6 = 3 33) If the value of tan 150 is 2 - √3, then what is the value of tan 150 cot 750 + tan 750 cot 150?
Answer: (a) 14 Explanation: tan 150 cot 750 + tan 750 cot 150 = ? tan 150 cot (900 - 150) + tan (900 - 150) cot 150 [as cot (900 - θ) = tan θ, and tan (900 - θ) = cot θ] = tan2 150 + cot2 150 …..(i) cot 150 = 1/tan 150 = 1 / 2-√3 = (1 / 2-√3) x (2+√3 / 2+√3) So, cot 150 = 2 + √3 So, on putting the values of cot 150 and tan 150 in equation (i), we will get = (2 - √3)2 + (2 + √3)2 = 4 + 3 - 2√3 + 4 + 3 + 2√3 = 14 34) Which of the following is the correct relation between A and B, if A = tan 110 . tan 290, and B = 2 cot 610 . cot 790?
Answer: (d) 2A = B Explanation: Given A = tan 110 . tan 290, and B = 2 cot 610 . cot 790 A / B = tan 110 . tan 290 / 2 cot 610 . cot 790 = [tan 110 . tan 290] / [2 cot (900 - 290) . cot (900 - 110)] = tan 110 . tan 290 / 2 tan 110 . tan 290 = 1/2 So, A/B = 1/2 Or, 2A = B 35) If sin θ x cos θ = 1/2, then what is the value of sin θ - cos θ?
Answer: (a) 0 Explanation: Given sin θ x cos θ = 1/2 On multiplying both sides by 2, we will get - 2 sin θ x cos θ = 1 sin 2θ = 1 (because sin 2θ = 2 sinθ cosθ) So, 2θ = 900 => θ = 450 Therefore, sin θ - cos θ = ? => sin 450 - cos 450 = 1/√2 - 1/√2 = 0 36) If the value of sin(θ + 300) is 3/√12, then what is the value of cos2 θ?
Answer: (a) 3/4 Explanation: Given sin (θ + 300) = 3/√12 It can be written as sin (θ + 300) = 3/2√3 Or, sin (θ + 300) = √3/2 => sin (θ + 300) = sin 600 => θ + 300 = 600 => θ = 300 On putting θ = 300, in cos2 θ, we will get cos2 300 = (√3/2)2 = 3/4 37) If the value of 4 cos2θ - 4√3 cos θ + 3 = 0, then what is the value of θ?
Answer: (b) 300 Explanation: In this example, we can find the value of θ by putting values given in options. It is the hit and trial approach. The value that satisfies the given equation will be considered as the value of θ. Given 4 cos2θ - 4√3 cos θ + 3 = 0 So, in option A θ = 600 is given, let's put it and see whether the equation will be satisfied or not - 4 cos2600 - 4√3 cos 600 + 3 = 0 4 x (1/2)2 - 4√3 x 1/2 + 3 = 0 => 4/4 - 4/2√3 + 3 = 0 => 4 - 4/2√3 = 0 => 4(1 - 1/2√3) = 0 (will not satisfy the equation) So, in option B, θ = 300 is given, let's put it and see whether the equation will be satisfied or not - 4 cos2300 - 4√3 cos 300 + 3 = 0 4 x (√3 /2)2 - 4√3 x √3/2 + 3 = 0 => 3 - 6 + 3 = 0 => 6 - 6 = 0 => 0 = 0 (Equation satisfied) So, option B is correct, and the value of θ is 300. 38) Which of the following is the correct value of cos2 550 + cos2 350 + sin2 650 + sin2 250?
Answer: (c) 2 Explanation: cos2 550 + cos2 350 + sin2 650 + sin2 250 => cos2 (900 - 350) + cos2 350 + sin2 650 + sin2 (900 - 650) => (sin2 350 + cos2 350) + (sin2 650 + cos2 650) => 1 + 1 = 2 39) If the value of tan 90 = p/q, then what is the value of sec2 810/ 1 + cot2 810?
Answer: (c) q2/p2 Explanation: sec2 810/ 1 + cot2 810 = sec2 810/ cosec2 810 = (1/cos2 810) / (1/sin2 810) = sin2 810 / cos2 810 = tan2 810 = tan2 (900 - 90) = cot2 90 = q2 / p2 40) If cot 450.sec 600 = A tan 300.sin 600, then which of the following is the correct value of A?
Answer: (a) 4 Explanation: Given cot 450.sec 600 = A tan 300.sin 600 So, 1 x 2 = A 1/√3 x √3/2 => 2 = A/2 So, A = 4 41) If the value of sec2 θ + tan2 θ = 7, then what is the value of θ?
Answer: (c) 600 Explanation: Given sec2 θ + tan2 θ = 7 => 1 + tan2 θ + tan2 θ = 7 => 2tan2 θ + 1 = 7 => 2tan2 θ = 6 => tan2 θ = 3 => tan θ = √3 Or θ = 600 We can also solve this question by using the hit and trial approach. We can directly check the values of θ given in options. The value that will satisfy the given condition will be the value of θ. 42) Which of the following is the correct value of (3 / 1+tan2 θ) + 2 sin2 θ + (1 / 1+cot2 θ)?
Answer: (a) 3 Explanation: (3 / 1+tan2 θ) + 2 sin2 θ + (1 / 1+cot2 θ) = ? According to the trigonometric identities, the given equation can be written as - = 3/sec2 θ + 2 sin2 θ + 1/cosec2 θ = 3cos2 θ + 2 sin2 θ + sin2 θ = 3cos2 θ + 3sin2 θ = 3(cos2 θ + sin2 θ) = 3 43) If the value of tan2 A = 1 + 2tan2 B, then what is the value of √2 cosA - cosB?
Answer: (a) 0 Explanation: Given tan2 A = 1 + 2tan2 B => sec2 A - 1 = 1 + 2 (sec2 B - 1) => sec2 A - 1 = 1 + 2 sec2 B - 2 => sec2 A - 1 = 2 sec2 B - 1 => 1/cos2 A = 2/cos2 B => cos2 B = 2cos2 A => or, cos B = √2 cos A => So, √2 cos A - cos B = 0 44) What will be the numerical value of (4 sec2 300 + cos2 600 - tan2 450) / (sin2 300 + cos2 300)?
Answer: (a) 55/12 Explanation: Given: (4 sec2 300 + cos2 600 - tan2 450) / (sin2 300 + cos2 300) We have to put the numerical values, = [4 (2/√3)2 + (½)2 - (1)2] / 1 => sec2 A - 1 = 1 + 2 (sec2 B - 1) => sec2 A - 1 = 1 + 2 sec2 B - 2 => sec2 A - 1 = 2 sec2 B - 1 => 1/cos2 A = 2/cos2 B => cos2 B = 2cos2 A => or, cos B = √2 cos A => So, √2 cos A - cos B = 0 45) If the value of tan A = 2, then what is the value of (cosec2 A - sec2 A) / (cosec2 A + sec2 A)?
Answer: (d) - 3/5 Explanation: Given: tan A = 2 (cosec2 A - sec2 A) / (cosec2 A + sec2 A) = ? On dividing above equation with cosec2 A, we will get - = (1 - tan2 A) / (1 + tan2 A) = (1 - 22) / (1 + 22) [because tan A = 2] = - 3/5 46) Which of the following is the correct value of (5/sec2 θ) + 3 sin2 θ + (2 / 1+cot2 θ)?
Answer: (c) 5 Explanation: (5 / sec2 θ) + 3 sin2 θ + (2 / 1+cot2 θ) = ? According to the trigonometric identities, the given equation can be written as - = 5cos2 θ + 3 sin2 θ + 2/cosec2 θ = 5cos2 θ + 3 sin2 θ + 2sin2 θ = 5cos2 θ + 5sin2 θ = 5(cos2 θ + sin2 θ) = 5 47) Which of the following is the correct numerical value of 5 tan2 A - 5 sec2 A + 1?
Answer: (d) -4 Explanation: 5 tan2 A - 5 sec2 A + 1 = ? = 5 (tan2 A - sec2 A) + 1 = 5 ((sin2 A/cos2 A) - (1/cos2 A)) + 1 = 5((sin2 A - 1) / cos2 A) + 1 = 5(- cos2 A / cos2 A) + 1 = - 5 + 1 = - 4 48) The value of cot 300/tan 600 is -
Answer: (c) 1 Explanation: tan 600 = √3, cot 300 = √3 So, cot 300/tan 600 = √3 / √3 = 1 49) If the value of tanP + secP = a, then what is the value of cosP?
Answer: (a) 2a/a2 + 1 Explanation: It is given that, tanP + secP = a ……(i) As we know, the trigonometric identity, sec2 P - tan2 P = 1 {we assume θ = P} So, we can apply the formula a2 - b2 = (a - b) (a + b) => (sec P - tan P) (sec P + tan P) = 1 => (sec P - tan P) x a = 1 => sec P - tan P = 1/a …..(ii) So, from equation (i) and (ii), we will get - 2sec P = a + 1/a sec P = a2+1 / 2a So, cos P = 2a / a2+1 [as sec P = 1/cosP] 50) Suppose cos θ + sin θ = √2 cos θ, then which of the following is the correct value of cos θ - sin θ?
Answer: (b) √2 sin θ Explanation: It is given that, cos θ + sin θ = √2 cos θ …..(i) On squaring both sides, we will get, (cos θ + sin θ)2 = (√2 cos θ)2 => cos2 θ + sin2 θ + 2 sin θ cos θ = 2 cos2 θ Or, 2cos2 θ - cos2 θ - sin2 θ = 2 sinθ cosθ => cos2 θ - sin2 θ = 2 sin θ cos θ => (cos θ + sin θ) (cos θ - sin θ) = 2 sin θ cos θ => (√2 cos θ) (cos θ - sin θ) = 2 sin θ cos θ [from equation (i)] => (cos θ - sin θ) = 2 sinθ cosθ / √2 cos θ = √2 sin θ
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